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111 & 113 WILLIAM STREET. NEW YORK.



PRACTICAL



MATHEMATICS,



DRAWING AND MENSURATION,



APPLIED TO



THE MECHANIC ARTS.



BY CHARLES DAVIES, LL.D.,

AUTHOR OF

FiHST LESSONS IN ARITHMETIC; ARITHMETIC J UNIVERSITY ARITHMETIC

ELEMENTARY ALGEBRA; ELEMENTARY GEOMETRY; ELEMENTS

OF SURVEYING ; ELEMENTS OF ANALYTICAL GEOMETRY ;

DESCRIPTIVE GEOMETRY ; SHADES, SHADOWS, AND

LINEAR PERSPECTIVE ; AND DIFFERENTIAL

AND INTEGRAL CALCULUS.



A. S. BARNES & COMPANY,

NEW YORK AND CHICAGO.

*f

1875.



DAVIES' MATHEMATICS.

fa® viit [email protected] ©@wssa 8

And Only Thorough and Complete Mathematical Series.



IIST THREE PAETS.



/. COMMON SCHOOL COURSE.

Davies' Primary Arithmetic— The fundamental principles displayed in

the Object Lessons.
Davies' Intellectual Arithmetic. — Referring all operations to the unit 1 as

the only tangible basis for logical development.
Davies' Elements of Written Arithmetic. A practical introduction

to the whole subject. Theory subordinated to Practice.
Davies' Practical Arithmetic* — The most successful combination of

Theory and Practice, clear, exact, brief, and comprehensive.

//. ACADEMIC COURSE.

Davies' University Arithme tic*— Treating the subject exhaustively as

a science, in a logical series of connected propositions.
Davies' Elementary Alyebra.*— A connecting link, conducting the pupil

easily from arithmetical processes to abstract analysis.
Davies' University Alyebra.*— For institutions desiring a more complete

but not the fullest course in pure Algebra.
Davies' Practical dlathem a tics. —The science practically applied to the

useful arts, as Drawing, Architecture, Surveying, Mechanics, etc.
Davies' Elementary Geometry. — The important principles in simple form,

but with all the exactness of vigorous reasoning.
Davies' Elements of Survey inr/.— Re-written in 1870. The simplest and

most practical presentation for youths of 12 to 16.

///. COLLEGIATE COURSE.

Davies' Bourdon's Alyebra.* — Embracing Sturm's Theorem, and a most
exhaustive and scholarly course.

Davies' University Alyebra.*— A shorter course than Bourdon, for Insti-
tutions have less time to give the subject.

Davies' Eeyendre's Geometry.— Acknowledged the only satisfactory trea-
tise of its grade. 300,000 copies have been sold.

Davies' Analytical Geometry and Calculus.— The shorter treatises,
combined in one volume, are more available for American courses of study.

Davies' Analytical Geometry. )The original compendiums, for those de-

Davies' Diff. & Int. Calculus. > siring to give full time to each branch.

Davies' Descriptive Geometry.— With application to Spherical Trigonome-
try, Spherical Projections, and Warped Surfaces.

Davies' Shades, Shadows, and Perspective— A succinct exposition of
the mathematical principles involved.

Davies' Science of Mathematics. — For teachers, embracing

I. Grammar of Arithmetic, TIT. Logic and Utility of Mathematics,
II. Outlines of Mathematics, IV. Mathematical Dictionary.



* Keys may be obtained from the Publishers by Teachers only.

Entered, according to Act of Congress, in the year 1852, by

CHARLES DAVIES,

In the Clerk's Office of the District Court of the United States for the Southern District

of New York.

FRAC. MATH.



PREFACE.



The design of the present work is to afford an ele-
mentary text-book of a practical character, adapted to the
wants of a community, where every day new demands
arise for the applications of science to the useful arts
There is little to be done, in such an undertaking, ex-
cept to collect, arrange, and simplify, and to adapt the
work, in all its parts, to the precise place which it is
intended to fill.

The introduction into our schools, within the last few
years, of the subjects of Natural Philosophy, Astronomy,
Mineralogy, Chemistry, and Drawing, has given rise to
a higher grade of elementary studies , and the extended
applications of the mechanic arts call for additional in-
formation among practical men.

To understand the most elementary treatise on Natu-
ral Philosophy, or the simplest work on the Mechanic
Arts, or even to make a plane drawing, some knowledge
of the principles of Geometry is indispensable ; and yet,
those in whose hands such works are generally placed,
or who are called upon to make plans in the mechanic
arts, feel that they have hardly time to go through with
a full course of exact demonstration.

The system of Geometry is a connected chain of rig-
orous logic. Every attempt to compress the reasoning,
by abridging it at the expense of accuracy, has been uni-
formly and strongly condemned.



IV PREFACE.

All the truths of Geometry necessary to carry out fully
the plan of the present work, are made accessible to the
general reader, without departing from the exactness of
the geometrical methods. This has been done by omit-
ting the demonstrations altogether, and relying for the
impression of each particular truth on a pointed question
and an illustration by a diagram. In this way, it is "be-
lieved that all the important properties of the geometrical
figures may be learned in a few weeks ; and after these
properties are developed in their practical applications,
the mind receives a conviction of their truth little short
of what is afforded by rigorous demonstration.

The work is divided into seven Books, and each book
is subdivided into sections.

In Book I. the properties of the geometrical figures
are explained by questions and illustrations.

In Book II. are explained the construction and uses
of the various scales, and also the construction of geo-
metrical figures. It is, as its title imports, Practical
Geometry.

Book III. treats of Drawing. — Section L, of the Ele
meats of the Art ; Section II., of Topographical Draw
ing ; and Section III., of Plan-Drawing.

Book IV. treats of Architecture, — explaining the dif-
ferent orders, both by descriptions and drawings.

Book V. contains the application of the principles of
Geometry to the mensuration of surfaces and solids. A
separate rule is given for each case, and the whole is
illustrated by numerous and appropriate examples.

Book VI. is the application of the preceding parts to
Artificers' Work. It contains full explanations of all the
scales and measures used by mechanics — the construc-
tion of these scales — the uses to which they are applied



PREFACE. V

— and specific rules for the calculations and computa-
tions which are necessary in practical operations.

Book VII. is an introduction to Mechanics. It ex-
plains the nature and properties of matter, the laws of
motion and equilibrium, and the principles of all the sim-
ple machines.

Book VIII. embraces a description of the Table of
Logarithms and their applications to many practical
questions ; also, many problems on the measurement of
heights and distances, and an article on the Strength of
Materials, adopted from Chambers' Educational Course.

From the above explanations, it will be seen that the
work is entirely practical in its objects and character
Many of the examples have been selected from a small
work somewhat similar in its object, recently published
in Dublin, by the Commissioners of National Education,
and some from a small French work of a similar charac
ter Others have been taken from Bonnycastle's Men
suration, and the Library of Useful Knowledge was
freely consulted in the preparation of Book VII. A
friend, Lt. Richard Smith, also furnished most of the
first and second sections of Book III. ; and the third
section was chiefly taken from an English work.

The author has indulged the hope that the present
work, together with his First Lessons in Arithmetic for
Beginners, his Arithmetic, Elementary Algebra, and Ele
mentary Geometry, will form an elementary course of
mathematical instruction adapted to the wants of Prac-
tical men, Academies and the higher grade of schools.

Fishkill Landing, ^September, 1852. *



DAVIES'

COURSE OF MATHEMATICS.



DAVIES' FIRST LESSONS IN ARITHMETIC— For beginner*

DAVIES' ARITHMETIC— Designed for the use of Academies and

Schoola

KEY TO DAVIES' ARITHMETIC.

DAVIES' UNIVERSITY ARITHMETIC— Embracing the Science
of Numbers, and their numerous applications.

KEY TO DAVIES' UNIVERSITY ARITHMETIC.

DAVIES' ELEMENTARY ALGEBRA— Being an Introduction to
the Science, and forming a connecting link between Arithmetic and
Algebra.

KEY TO DAVIES' ELEMENTARY ALGEBRA.

DAVIES' ELEMENTARY GEOMETRY— This work embraces the
elementary principles of Geometry. The reasoning is plain and con-
cise, but at the same time strictly rigorous.

DAVIES' PRACTICAL MATHEMATICS*, WITH DRAWING AND

MENSURATION— Applied to the Mechanic Arts.

DAVIES' BOURDON'S ALGEBRA— Including Sturms' Theorem,- -
Being an Abridgment of the work of M. Bourdon, with the addition of
practical examples.

DAVIES' LEGENDRE'S GEOMETRY and TRIGONOMETRY

— Being an Abridgment of the work of M. Legendre, with the addition
of a Treatise on Mensuration of Planes and Solids, and a Table of
Logarithms and Logarithmic Sines.

DAVIES' SURVEYING— With a description and plates of the Theod-
olite, Compass, Plane-Table, and Level : also, Maps of the Topo-
graphical Signs adopted by the Engineer Department — an explana-
tion of the method of surveying the Public Lands, and an Elementary
Treatise on Navigation.

DAVIES' ANALYTICAL GEOMETRY— Embracing the Equa-
tions of the Point and Straight Line — of the Conic Sections — of
the Line and Plane in Space — also, the discussion of the General
Equation of the second degree, and of Surfaces of the second order.

DAVIES' DESCRIPTIVE GEOMETRY,— With its application to
Spherical Projections.

DAVIES' SHADOWS* and LINEAR PERSPECTIVE.

DAVIES' DIFFERENTIAL and INTEGRAL CALCULUS



CONTENTS.



BOOK I.— SECTION I.



_•""



Page



Of Lines and Angles 13

Of Parallel Lines — Oblique Lines 14

Of Horizontal Lines — Vertical Lines 14

Of Angles formed by Straight Lines — By Curves 15

Of the Right Angle — Acute Angle — Obtuse Angle 15—16

Two Lines intersecting each other 16 — 11

Parallels cut by a third Line — Oblique Lines 17

Ol the Circle, and Measurement of Angles 17

Degrees in a Right Angle — Quadrant 18

Sum of the Angles on the same Side of a Line 19

Sum of the Angles about a Point 19

SECTION II.

Plane Figures 20

Different Kinds of Polygons 21

Different Kinds of Quadrilaterals 20—22

Diagonal of a Quadrilateral 23

Square on the Hypothenuse of a Right-angled Triangle 23

SECTION III.

Of the Circle, and Lines of the Circle 24

Radius of the Circle — Diameter of the Circle 24

Arc — Chord — Segment — Sector 25

Angle at the Centre — At the Circumference 26

Angle in a Segment — Secant — Tangent 26

Figure inscribed in a Circle — Figure circumscribed about it 27

Measure of an Angle at the Centre — At the Circumference . . . 27 — 28

Sum of the Angles of a Triangle— Chords of ihc Circle 28 — 29



V1I1 CONTENTS.

BOOK II.— SECTION I.

Tape

Practical Geometry 30

Description of Dividers, and Uso 30 — 31

Ruler and Triangle, and Use 32 — 33

Scale of Equal Parts, and Use 33 — 34

Diagonal Scale of Equal Parts, and Uso 36—37

Scale of Chords, and Use 38

Protractor, and Use 38

Gunter's Scale 39

Practical Problems.. 40—50

Questions for Practice 50 — 52

BOOK III.— SECTION I.

Drawing in General 53

Illustration of Form— Of Shade and Shadow 53—60

Manner of using the Pencil 60 — 61

General Remarks 61 — 63

SECTION II.

Topographical Drawing..... 63

Description of Topographical Drawing 63

Explanation of the Figures and Signs 64 — 70

SECTION III.

Plan Drawing 70

Geometrical Drawings — Denned 70

Horizontal Plane — Denned 70

Vertical Plane — Defined 71

Plan — Denned 71

Illustrations of Plan 71—78

Sections 78—82

The Elevation 82—86

Remarks on Elevations 86* — 88

Oblique Elevations 88 — 95

General Remarks 95 — 96

BOOK IV.— SECTION I.

Of Architecture 97

Definition of Architecture — How divided 97

Elements of Architecture — Mouldings 97 — 100



CONTENTS. IX

SECTION IL

Page

Orders of Architecture — Their Parts 102 — 104

Tuscan Order : 105

Doric Order .. 105

Ionic Order , 107

Corinthian Order \. ]07

BOOK V.— SECTION I.

Mensuration of Surfaces 109

Unit of Length, or Linear Unit 109

Unit of Surface, or Superficial Unit 109

Meaning of the term Rectangle 110

Denominations in which Areas are computed 1 12 — 1 14

Area of the Triangle ' 114 — 117

Properties of the Right-angled Triangle 117 — 119

Area of the Square 119—120

Area of the Parallelogram 120 — 121

Area of the Trapezoid 121—122^

Area of the Quadrilateral — Of an Irregular Figure 122 — 1 25

Areas and Properties of Polygons 125 — 132

Of the Circle— Area and Properties 132—143

Of Circular Rings 143—144

Area of the Ellipse 144—145

SECTION II.

Mensuration of Solids 145

Definition of a Solid— Different Kinds 145—147

Content of Solids— Unit of Solidity— Table 147—149

Of the Prism 149 — 152

Of the Pyramid 152—157

Of the Frustum of a Pyramid 157 — 159

Of the Cylinder 159—163

Of the Cone 163—166

Of the Frustum of a Cone 167—169

Of the Sphere 169—173

Of Spherical Zones „ 174

Of Spherical Segments 1?4 — 176

Of the Spheroid 176—178

Of Cylindrical Rings 178—179

Of tho Five Regular Solids 179—183

1*
\



X CONTENTS.

BOOK VI.

Pag»

Artificers' Work , 184

SECTION I.

Of Measures 184

Carpenters' Rule — Description and Uses 184 — 187

To multiply Numbers by the Carpenters' Rule 187 — 190

To find the Content of a Piece of Timber by the Carpenters'

Rule 190

Table for Board Measure 191

Board Measure 192

SECTION II.

Of Timber Measure 193

To find the Area of a Plank 193—194

To cut a given Area from a Plank 195

To find the solid Content of a square Piece of Timber 195 — 197

To cut off* a given Solidity from a Piece of Timber 197

To find the Solidity of round Timber 198

Of Logs for Sawing 199

To find the Number of Feet of Boards which can be sawed

from a Log 200

SECTION III.

Bricklayers' Work 201

How Artificers' Work is computed 201

Dimensions of Brick 202

To find the Number of Bricks necessary to build a given Wall 202

Of Cisterns 204

To find the Content of a Cistern in Hogsheads 204

Having the Height of a Cistern, to find its Diameter that it

may contain a given Quantity of Water 205

Having the Diameter, to find the Height 205

SECTION IV.

Masons' Work 20G

SECTION V.

Carpenters' and Joiners' Work 207

Of Bins for Grain 203



CONTENTS. XI

Page

To fiud the Number of cubic Feet in any Number of Bushels 208
To find the Number of Bushels which a Bin of a given Size

will hold 208

To find the Dimensions of a Bin which shall contain a given

Number of Bushels.... 209

SECTION VI.

Slaters' and Tilers' Work 210

SECTION VII.

Plasterers' Work 210

To find the Area of a Cornice 211

SECTION VIII.

Painters' Work 212

SECTION IX.

Pavers' Work 212

SECTION X.

Plumbers' Work 213

BOOK VII.

Introduction to Mechanics 215

SECTION I.

Of Matter and Bodies 215

Matter— Defined 215

Body— Denned 215

Space — Defined 215

Of the Properties of Bodies 215

Impenetrability — Defined 215

Extension — Defined 2l(i

Figure— Defined . 2lfi

Divisibility— Defined 216

Inertia — Defined 217

Atoms— Defined 217

Attraction of Cohesion 217

Attraction of Gravitation 218

Weight— Defined 219



Xll CONTENTS.

SECTION II.

Page

Laws of Motion, and Centre of Gravity 219

Motion — Defined 219

Force or Power — Defined 219

Velocity— Defined 219

Momentum — Defined 221

Action and Reaction — Defined 221

Centre of Gravity— Defined 222

SECTION III.

Of the Mechanical Powers 224

General Principles..... 224

Lever— Different Kinds 224—227

Pulley 227—229

Wheel and Axle 230

Inclined Plane 231

Wedge— Screw 232

General Remarks 233

SECTION IV.

Of Specific Gravity 234

Specific Gravity — Defined 234

When a Body is specifically heavier or lighter than another 234

Density— Defined 234

To find the Specific Gravity of a Body heavier than Water 236

To find the Specific Gravity of a Body lighter than Water 237

To find the Specific Gravity of Fluids 238

Table of Specific Gravities 239

To find the Solidity of a Body when its Weight and Specific

Gravity are known 240

BOOK VIII.

Applications of Mathematics 241

Logarithms 241- 253

Applications to Heights and Distances 253- -271

Strength of Materials 271 —296



GEOMETRY.



BOOK I.

SECTION I.

OF LINES AND ANGLES.

1. What is a line?

A Line is length, without breadth or thickness.

2. What are the extremities of a line called ?

The Extremities of a Line are called points ; and any
place between the extremities, is also called a point.

3. What is a straight line ?

A Straight Line, is the shortest dis-
tance from one point to another. Thus,
AB is a straight line, and the shortest
distance from A to B.

4. What is a curve line ?

A Curve Line, is one which changes
its direction at every point. Thus, .
ABC is a curve line.

5. What does the word line mean ?

The word Line, when used by itself, means a straight
line; and the word Curve, means a curve line.



14 BOOK I. SECTION I.

6. What is a surface ?

A Surface is that which has length and breadth, without
height or thickness.

7. What is a plane, or plane surface ?

A Plane is that which lies even throughout its whole ex
tent, and with which a straight line, laid in any direction,
will exactly coincide.

8. When are lines said to be parallel?

Two straight lines are said to be paral-
lel when they are at the same distance —

from each other at every point. Parallel

^nes will never meet each other.

9. When are two curves said to be parallel?
Two curves are said to be parallel or

concentric, when they are at the same dis-
tance from each other. Parallel curves
will not meet each other.

10. What are oblique lines?

Oblique lines are those which ap-
proach each other, and meet if suffi-
ciently prolonged.

11. What are horizontal lines?

Lines which are parallel to the horizon, or to the water
level, are called Horizontal Lines. Thus, the eaves of a
house are horizontal.

12. What are vertical lines?

All plumb lines are called Vertical Lines. Thus, trees
and plants grow in vertical lines.

13. What is an angle? How is it read?

An Angle is the opening or inclination of two lines which



OF LINES AND ANGLES.



^



15




meet each other in a point. Thus the
lines AC, AB, form an angle at the point
A. The lines AC, and AB, are called
the sides of the angle, and their intersec-
tion A, the vertex.

The angle is generally read by placing the letter at the
vertex in the middle : thus, we say the angle CAB. We
may, however, say simply, the angle A.



14. May angles be formed by curved
lines 1

Yes, either by two curves, CA, BA A
forming the angle A, called a curvilinear
angle :



Or, by two curves AC, AB, forming
the angle A:



£



Or, by a straight line and curve, which
is called a mixtilinear angle.




D



15. When is one line said to be perpendicular to another ?

One line is perpendicular to another,
when it inclines no more to the one
side than to the other. Thus, the line
DB is perpendicular to AC, and the
angle DBA is equal to DBC.



A



~W



16



BOOK I. SECTION I.



16. What are the angles called?

When two lines are perpendicular to
each other, the angles which they form
are called right angles. Thus, DBA
and DBC are right angles. Hence, all
right angles are equal to each other.



B



~fi



17. What is an acute angle?

An acute angle is less than a right
angle. Thus, DBC is an acute angle.

18. What is an obtuse angle?

An obtuse angle is greater than a
right angle. Thus, EBC is an obtuse
angle.



D
~C



B



19. If two lines intersect each other, what follows?


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Online LibraryCharles DaviesPractical mathematics, with drawing and mensuration, applied to the mechanic arts → online text (page 1 of 20)